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A number is called Rational if it can be expressed in the form p/q where p and q are integers (q > 0). It includes all natural, whole number and integers.
Example: 1/2, 4/3, 5/7,1 etc.
All the positive integers from 1, 2, 3,……, ∞.
All the natural numbers including zero are called Whole Numbers.
Integers
All negative and positive numbers including zero are called Integers.
This shows that the operation of any two same types of numbers is also the same type or not.
a. Whole Numbers
If p and q are two whole numbers then
Whole number
p + q will also be the whole number.
p – q will not always be a whole number.
pq will also be the whole number.
p ÷ q will not always be a whole number.
Example
6 + 0 = 6
8 – 10 = – 2
3 × 5 = 15
3 ÷ 5 = 3/5
Closed or Not
Closed
Not closed
b. Integers
If p and q are two integers then
p+q will also be an integer.
p-q will also be an integer.
pq will also be an integer.
p ÷ q will not always be an integer.
- 3 + 2 = – 1
5 – 7 = – 2
- 5 × 8 = – 40
- 5 ÷ 7 = – 5/7
Closed or not
c. Rational Numbers
If p and q are two rational numbers then
Rational Numbers
p + q will also be a rational number.
p – q will also be a rational number.
pq will also be a rational number.
p ÷ q will not always be a rational number
p ÷ 0
= not defined
This shows that the position of numbers does not matter i.e. if you swap the positions of the numbers then also the result will be the same.
p + q = q + p
p – q ≠ q – p
p × q = q × p
p ÷ q ≠ q ÷ p
3 + 2 = 2 + 3
8 –10 ≠ 10 – 8 – 2 ≠ 2
3 × 5 = 5 × 3
3 ÷ 5 ≠ 5 ÷ 3
Commutative
yes
No
True
5 – 7 = – 7 – (5)
- 5 × 8 = 8 × (–5)
- 5 ÷ 7 ≠ 7 ÷ (-5)
Rational numbers
p –q ≠ q - p
This shows that the grouping of numbers does not matter i.e. we can use operations on any two numbers first and the result will be the same.
If p, q and r are three whole numbers then
p + (q + r) = (p + q) + r
p – (q – r) = (p – q) – r
p × (q × r) = (p × q) × r
p ÷ (q ÷ r) ≠ (p ÷ q) ÷ r
3 + (2 + 5) = (3 + 2) + 5
8 – (10 – 2) ≠ (8 -10) – 2
3 × (5 × 2) = (3 × 5) × 2
10 ÷ (5 ÷ 1) ≠ (10 ÷ 5) ÷ 1
Associative
If p, q and r are three integers then
Addition
(– 6) + [(– 4)+(–5)] = [(– 6) +(– 4)] + (–5)
Yes
Subtraction
5 – (7 – 3) ≠ (5 – 7) – 3
Multiplication
(– 4) × [(– 8) ×(–5)] = [(– 4) × (– 8)] × (–5)
Division
[(–10) ÷ 2] ÷ (–5) ≠ (–10) ÷ [2 ÷ (– 5)]
If p, q and r are three rational numbers then
Zero is the additive identity for whole numbers, integers and rational numbers.
a + 0 = 0 + a = a
Addition of zero to whole number
2 + 0 = 0 + 2 = 2
Integer
b + 0 = 0 + b = b
Addition of zero to an integer
False
Rational number
c + 0 = 0 + c = c
Addition of zero to a rational number
2/5 + 0 = 0 + 2/5 = 2/5
One is the multiplicative identity for whole numbers, integers and rational numbers.
a ×1 = a
Multiplication of one to the whole number
5 × 1 = 5
b × 1= b
Multiplication of one to an integer
- 5 × 1 = – 5
Rational Number
c × 1= c
Multiplication of one to a rational number
a +(- a) = 0
Where a is a whole number
5 + (-5) = 0
b +(- b) = 0
Where b is an integer
c + (-c) = 0
Where c is a rational number
The multiplicative inverse of any rational number
The reciprocal of 4/5 is 5/4.
This shows that for all rational numbers p, q and r
1. p(q + r) = pq + pr
2. p(q – r) = pq – pr
Check the distributive property of the three rational numbers 4/7,-( 2)/3 and 1/2.
Solution
Let’s find the value of
This shows that
On the number line, we can represent the Natural numbers, whole numbers and integers as follows
Rational Numbers can be represented as follows
There could be n number of rational numbers between two rational numbers. There are two methods to find rational numbers between two rational numbers.
Method 1
We have to find the equivalent fraction of the given rational numbers and write the rational numbers which come in between these numbers. These numbers are the required rational numbers.
Find the rational number between 1/10 and 2/10.
As we can see that there are no visible rational numbers between these two numbers. So we need to write the equivalent fraction.
2/10 = 20/100((multiply the numerator and denominator by 10)
Hence, 2/100, 3/100, 4/100……19/100 are all the rational numbers between 1/10 and 2/10.
Method 2
We have to find the mean (average) of the two given rational numbers and the mean is the required rational number.
To find mean we have to divide the sum of two rational numbers by 2.
3/20 is the required rational numbers and we can find more by continuing the same process with the old and the new rational number.
Remark: 1. This shows that if p and q are two rational numbers then (p + q)/2 is a rational number between p and q so that
p < (p + q)/2 < q.
2. There are infinite rational numbers between any two rational numbers.
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